Algebra and Applications 1. Abdenacer Makhlouf
rel="nofollow" href="#fb3_img_img_4253a718-ed69-56d1-a0f3-d890f6f87954.jpg" alt="image"/>, Q(n)(+) are of the type A(+), where A is a simple finite dimensional associative superalgebra.
EXAMPLE 1.27.– Every right module over A gives rise to a one-sided Jordan bimodule over A(+).
Suppose now that the superalgebra A is equipped with an involution ∗.
EXAMPLE 1.28.– The subspaces H(A, ∗) and K(A, ∗) become Jordan A(+)- bimodules with the actions:
1 1) h ○ a = ha + a∗h;
2 2) h ○ a = ha∗ + ah;
3 3) k ○ a = ka + a∗k;
4 4) k ○ a = ka∗ + ak;
where h ∈ H(A, ∗), k ∈ K(A, ∗), a ∈ A.
The associative superalgebras Mm+n(F), where both m, n are odd, and Q(n), are not equipped with an involution, but they are equipped with a pseudoinvolution.
DEFINITION 1.20.– A graded linear map ∗ : A → A is called a pseudoinvolution if (ab)∗ = (–1)|a|∙|b|b∗a∗, (a∗)∗ = (–1)|a| a for arbitrary elements
EXAMPLE 1.29.– The mapping
EXAMPLE 1.30.– The mapping
Replacing the involution ∗ in example 1.28 with the pseudoinvolutions of examples 1.29 and 1.30, we get unital Jordan bimodules over Mm+n(F), where m, n are odd, and over Q(n)(+).
THEOREM 1.7 (see Martin and Piard (1992), Martínez et al. (2010)).– An arbitrary irreducible Jordan bimodule over
The exceptional Jordan superalgebra K10 has rank 3. Jordan bimodules over K10 have been classified by Shtern (1987).
THEOREM 1.8 (Shtern (1987)).– All Jordan bimodules over K10 are completely reducible. The only irreducible Jordan bimodules over K10 are the regular bimodule and its opposite.
1.7.2. Superalgebras of rank ≤ 2
If J is a Jordan superalgebra of rank ≤ 2, then, generally speaking, it is no longer true that its universal multiplicative algebra is finite dimensional and that any Jordan bimodule is completely reducible.
1.7.2(a) In the case J = Q(2)(+), however, it is true (see Martínez et al. (2010)). The universal multiplicative enveloping algebra U(Q(2)(+)) is finite dimensional and semisimple and the description of irreducible Jordan bimodules is similar to that of Q(n)(+), n ≥ 3.
1.7.2(b) Let us discuss bimodules over Kantor superalgebras. Recall that the Kantor superalgebras Kan(n) are Kantor doubles of the Grassmann superalgebras G(n), n ≥ 1, with respect to the Poisson bracket
Let Kan(n) = G(n) + G(n)v. The Grassmann superalgebra G(n) is embeddable in the associative commutative superalgebra A = F [t, ξ1,…, ξn] = F [t] ⊗F G(n).
For an arbitrary scalar α ∈ F, the Poisson bracket [ , ] extends to the Jordan bracket on A defined by [t, ξi] = 0, [ξi, ξj] = –δij, [ξi, 1] = 0, [t, 1] = αt. The Kantor double Kan(n) = G(n) + G(n)v embeds in the Kantor double Kan(A, [ , ]) = A + Av. The subspace V (α) = tG(n) + tG(n)v is an irreducible unital Jordan bimodule over K(n). The square of the multiplication operator by the element v acts on V (α) as the scalar multiplication by α.
The simple superalgebras Kan(n), n ≥ 2 are exceptional (see Martínez et al. (2001)). Therefore, they do not have non-zero one-sided Jordan bimodules.
THEOREM 1.9 (Stern (1995), Martínez and Zelmanov (2009), Solarte and Shestakov (2016)).– Every finite dimensional irreducible Jordan bimodule over Kan(n), n ≥ 2 is isomorphic to V (α) or V (α)op, α ∈ F.
In Solarte and Shestakov (2016), the theorem above was proved for algebras over a field of characteristic p > 2.
1.7.2(c) Jordan superalgebras of a superform. Let
Let Cl(m) be the Clifford algebra of the restriction of the form 〈 , 〉 to
be the simple Weyl algebra.
Then the tensor product S = Cl(m) ⊗F Wn is the universal associative enveloping superalgebra of the Jordan superalgebra J = V + F ∙ 1.
Since the algebra Wn, n ≥ 1 is infinite dimensional, it follows that the superalgebra J does not have non-zero finite dimensional one-sided Jordan bimodules unless n = 0.
Consider in the algebra